There are 160 People in a society. Out of them, 130 has own profile on Twitter, 120 has own profile on Facebook, 90 has own profile on Instagram, 150 has own profile on Linkedin. What is the minimum number of People in society that have Profiles on all these 4 Social Media? Assume each Person in society has profile on at least one of these Social Media.
(A) 10
(B) 20
(C) 90
(D) 120


Answer: (A)

Explanation: Let A,B,C,D ⊂ {1, 2, …,160} be the four sets, with

|A| = 130, |B| = 120, |C| = 90, |D| = 150 

Then we want the minimum size of (A∩B∩C∩D). Combining the fact that,

|A∩B∩C∩D| = 160 − |A’∪B’∪C’∪D’| 

where A’ refers to A complement, along with the fact that for any sets

|X∪Y|≤|Y|+|X| 

we see that,

|A∩B∩C∩D| ≥ 160 − |A’|−|B’|−|C’|−|D’| 
= 160 - 30 - 40 - 70 - 10
= 10 

So, option (A) is correct


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